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Dual Complex Adjoint Matrix: Applications in Dual Quaternion Research

Yongjun Chen, Liping Zhang

TL;DR

This work introduces the dual complex adjoint matrix $\\mathcal{J}$ for dual quaternion matrices and develops its theory to transform dual-quaternion problems into the complex domain. It establishes canonical standard right eigenvalues for dual quaternion matrices, and leverages the adjoint to obtain direct solutions for Hand-Eye calibration models $AX=XB$ and $AX=YB$, while converting dual quaternion linear systems into dual complex linear systems. The authors then integrate $\\mathcal{J}$ into a Rayleigh quotient iteration, yielding a faster algorithm with substantial reductions in floating-point operations, particularly when the standard part is positive definite. Numerical experiments validate Hand-Eye calibration solutions and demonstrate notable speedups in eigenvalue computations for Laplacian matrices in formation-control contexts, highlighting the practical impact of this dual-complex approach.

Abstract

Dual quaternions and dual quaternion matrices have garnered widespread applications in robotic research, and its spectral theory has been extensively studied in recent years. This paper introduces the novel concept of the dual complex adjoint matrices for dual quaternion matrices. We delve into exploring the properties of this matrix, utilizing it to study eigenvalues of dual quaternion matrices and defining the concept of standard right eigenvalues. Notably, we leverage the properties of the dual complex adjoint matrix to devise a direct solution to the Hand-Eye calibration problem. Additionally, we apply this matrix to solve dual quaternion linear equations systems, thereby advancing the Rayleigh quotient iteration method for computing eigenvalues of dual quaternion Hermitian matrices, enhancing its computational efficiency. Numerical experiments have validated the correctness of our proposed method in solving the Hand-Eye calibration problem and demonstrated the effectiveness in improving the Rayleigh quotient iteration method, underscoring the promising potential of dual complex adjoint matrices in tackling dual quaternion-related challenges.

Dual Complex Adjoint Matrix: Applications in Dual Quaternion Research

TL;DR

This work introduces the dual complex adjoint matrix for dual quaternion matrices and develops its theory to transform dual-quaternion problems into the complex domain. It establishes canonical standard right eigenvalues for dual quaternion matrices, and leverages the adjoint to obtain direct solutions for Hand-Eye calibration models and , while converting dual quaternion linear systems into dual complex linear systems. The authors then integrate into a Rayleigh quotient iteration, yielding a faster algorithm with substantial reductions in floating-point operations, particularly when the standard part is positive definite. Numerical experiments validate Hand-Eye calibration solutions and demonstrate notable speedups in eigenvalue computations for Laplacian matrices in formation-control contexts, highlighting the practical impact of this dual-complex approach.

Abstract

Dual quaternions and dual quaternion matrices have garnered widespread applications in robotic research, and its spectral theory has been extensively studied in recent years. This paper introduces the novel concept of the dual complex adjoint matrices for dual quaternion matrices. We delve into exploring the properties of this matrix, utilizing it to study eigenvalues of dual quaternion matrices and defining the concept of standard right eigenvalues. Notably, we leverage the properties of the dual complex adjoint matrix to devise a direct solution to the Hand-Eye calibration problem. Additionally, we apply this matrix to solve dual quaternion linear equations systems, thereby advancing the Rayleigh quotient iteration method for computing eigenvalues of dual quaternion Hermitian matrices, enhancing its computational efficiency. Numerical experiments have validated the correctness of our proposed method in solving the Hand-Eye calibration problem and demonstrated the effectiveness in improving the Rayleigh quotient iteration method, underscoring the promising potential of dual complex adjoint matrices in tackling dual quaternion-related challenges.
Paper Structure (22 sections, 11 theorems, 97 equations, 1 table, 2 algorithms)

This paper contains 22 sections, 11 theorems, 97 equations, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. dual quaternion
  4. Dual number and dual complex number
  5. quaternion
  6. dual quaternion
  7. Dual quaternion matrix and eigenvalues of dual quaternion matrix
  8. Dual quaternion matrix
  9. Eigenvalues of dual quaternion matrices
  10. Dual Complex Adjoint Matrix and its Applications
  11. Dual complex adjoint matrix
  12. Standard eigenvalues of dual quaternion matrices
  13. Application of dual complex adjoint matrix in Hand-Eye calibration problem
  14. AX=XB Hand-Eye calibration problem
  15. AX=YB Hand-Eye calibration problem
  16. ...and 7 more sections

Key Result

Lemma 1

Let $\tilde{\mathbf A}= A_1+ A_2 j\in \mathbb Q^{n\times n}$, $\tilde{\mathbf x}=\mathbf x_1+\mathbf x_2 j\in \mathbb Q^{n\times 1}$, $\lambda\in \mathbb C$, where $A_1, A_2\in\mathbb C^{n\times n}$, $\mathbf x_1,\mathbf x_2\in\mathbb C^{n\times 1}$, then is equivalent to

Theorems & Definitions (34)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 24 more